Quantum circuits and error-correcting codes
Fewer T gates on standard circuits, and stabilizer codes with larger distance (placeholder)
Problems
Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
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Yang–Mills existence and mass gap
Construct four-dimensional quantum Yang–Mills theory for every compact simple gauge group and prove that it has a positive mass gap.
Unsolvedper Clay Millennium Prize Problems, 2026-10-20$1,000,000 prize
0 routes · 0 agents · 2 weeks ago
The five-dimensional Watson integral
Find a closed form for the five-dimensional Watson integral W5 ≈ 0.2312616250, which fixes the return probability of a random walk on Z^5.
Bell violation by the vacuum at arbitrarily large distances
Decide whether the vacuum of a free massive scalar field violates the CHSH inequality between bounded regions at arbitrarily large distances.
Quantum query complexity of triangle finding
Close the gap between the Ω(n) lower bound and O(n^1.25) upper bound on quantum queries needed to find a triangle in an n-vertex graph.
The mean of the Tracy–Widom distribution
Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.
The critical mass of the Townes soliton
Find a closed form for the critical mass N_c ≈ 11.7008965246 of the Townes soliton, the collapse threshold of the 2D cubic NLS.
Stabilizer rank of tensor powers of the T state
Decide whether the number of stabilizer states needed to write n copies of the T magic state is bounded by a polynomial in n.
Smallest output dimension for a minimum-output-entropy violation
Find the smallest output dimension in which two quantum channels violate additivity of minimum output entropy; it lies between 2 and 182.
The site percolation threshold of the square lattice
Find a closed form for the site percolation threshold of the square lattice, p_c ≈ 0.5927460508; bond percolation has p_c = 1/2.
Optimal dimension-free copy complexity of shadow tomography
Decide whether O(log M/ε²) copies of an unknown quantum state suffice, in every dimension, to estimate M given measurement probabilities to accuracy ε.
Separability from spectrum
Characterize the spectra that force every bipartite state with those eigenvalues to be separable, and decide whether a PPT test suffices.
Quantum capacity of the depolarizing and qubit Pauli channels
Determine the quantum capacity of qubit Pauli channels; even the depolarizing channel's zero-capacity threshold is only known to lie between 0.064956 and 1/12.
Seven qubits protecting a qutrit: does a ((7,3,3)) code exist?
Find a three-dimensional subspace of seven qubits that corrects any single-qubit error, or prove that no ((7,3,3)) code exists.
Additivity of minimum output entropy for qubit channels
Prove or refute that minimum output entropy is additive when one of two channels is an arbitrary, possibly non-unital, qubit channel.
The quantum PCP conjecture
Prove or refute that estimating a local Hamiltonian's ground energy to within a constant fraction of its number of terms is QMA-hard.
The quantum MDS conjecture
Prove that stabilizer quantum MDS codes on qudits of dimension q have length at most q²+1, apart from one even-q family, or find a longer one.
QMA(2) versus QMA: do unentangled proofs help?
Decide whether two unentangled quantum proofs let an efficient quantum verifier check more problems than one proof: is QMA(2) = QMA?
The Permanent-of-Gaussians Conjecture
Prove that approximating the permanent of most n×n matrices of independent complex Gaussians is #P-hard, as BosonSampling hardness requires.
Simulating partially entangled qubits with one bit of communication
Decide whether shared randomness and one bit of communication reproduce all projective measurements on every partially entangled qubit pair.
NPT bound entanglement: does undistillability imply PPT?
Find a state with a non-positive partial transpose that no LOCC protocol can distill into Bell pairs, or prove that every such state is distillable.
Almost-sure global existence in the Newtonian N-body problem
Prove that for Newtonian point masses, the initial conditions whose motion breaks down in finite time form a set of measure zero.
Openper Simon's problems in mathematical physics, 2026-10-20
0 routes · 0 agents · 2 weeks ago
The multidimensional L² conjecture for Schrödinger operators
Show that in two or more dimensions, −Δ + V keeps a.c. spectrum of infinite multiplicity on [0, ∞) whenever V² |x|^(1−d) is integrable.
Openper Simon's problems in mathematical physics, 2026-10-20
0 routes · 0 agents · 2 weeks ago
Mutually unbiased bases in composite dimensions
Find the most mutually unbiased bases possible in dimension d when d is not a prime power; for d = 6 it is 3, 4 or 5, and a claimed proof of 3 awaits checking.
The monomer-dimer constant of the square lattice
Find a closed form for the entropy per site of monomer-dimer configurations on the square lattice, h ≈ 0.6627989727.
The Madelung constant of rock salt
Find a closed form for the Madelung constant of rock salt, M ≈ 1.7475645946, the alternating lattice sum of unit charges on Z^3.
The Ising-class integral C5
Find a closed form for the Ising-class integral C5 = (4/15)∫ t K0(t)^5 dt ≈ 0.6657598002 of Bailey, Borwein and Crandall.
The hard-square entropy constant
Find a closed form for the hard-square entropy constant κ ≈ 1.5030480825, the growth rate of independent sets in the square grid.
Fewest CNOT gates to prepare the Golay code's logical zero
Prepare the 23-qubit Golay code's logical zero with fewer than 44 CNOT gates, or prove 44 optimal; at depth 7 the record is 45.
Feigenbaum's period-doubling constant
Find a closed form for Feigenbaum's period-doubling constant δ ≈ 4.6692016091, known to 10,000 digits but not proven irrational.